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Full text of "Treatise On Analysis Vol-Ii"

10 SPACES WITH OPERATORS. ORBIT SPACES 49 Let n(x) and n(y) be distinct points of E/G. If E/G is Hausdorff, there exist saturated open sets V, W in E such that V n W = 0 and x e V and y e W. It is clear that the open set V x W in E x E contains (x, y) and does not meet R. Hence R is closed in E x E. Conversely, if R is closed in E x E, there exists an open neighborhood S of x and an open neighborhood T of y in E such that (S x T) n R = 0. By (12.10.6), 7i(S) and n(T) are neighborhoods of n(x) and n(y), respectively. If they intersected, there would exist se S and teT belonging to the same orbit, which means that (s, t)eR; and this is absurd. Hence E/G is Hausdorff. The last assertion follows from (12.3.4) and the continuity of n. (12.10.9) Let E be a metrizable space, G a topological group acting continu- ously on E, and n : E -»E/G the canonical mapping. Suppose that E/G is metrizable. Then: (i) ifE is separable, E/G is separable. (ii) ifE is locally compact (resp: compact), E/G is locally compact (resp. compact); (iii) if E is locally compact and K is any compact subset of E/G, there exists a compact subset LofE such that K = n(L). If D is a denumerable dense subset of E, then n(D) is dense in E/G (3.11.4(d)). This proves (i). If V is a compact neighborhood of xe E, then 7t(V) is a compact neighborhood of n(x) in E/G ((3.17.9) and (12.10.6)); hence (ii). Finally, for each z e K, let V(z) be a compact neighborhood of a point of TT~I(Z) in E, so that n(V(z)) is a compact neighborhood of z. There exist a finite number of points zt e K such that the rc(Vfo)) cover K. Let Lj be the compact set (J V(z,) in E. We have K c 71(1^), hence the set L = Lj n ^"'(K) is compact (because it is closed in Lj : (3.11.4) and (3.17.3)) and we have 7i(L) = K. (12.10.10) Let E be a locally compact, separable metrizable space and G a topological group acting continuously on E. Let n:E~+ E/G be the canonical mapping. Suppose that (1) E/G is Hausdorff; (2) for each xeE there exists a compact subset K(x) of E, containing x and such that the restriction of n to K(x) is injective and 7r(K(^:)) is a neighborhood ofn(x). Then E/G is metrizable (and therefore, by (12.10.9), locally compact and separable). We shall first show that there exists a sequence (Kw) of compact subsets of E such that the restriction of n to Kw is injective and the interiors of the sets n(Kn) cover E/G. By (3.18.3) there exists an increasing sequence (Aw) of com- pact subsets of E, whose union is E. For each z e n(An), let x be a point offt action) of G on E is a